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 A196669 The Chebyshev primes of index 3. 5
 11, 19, 29, 61, 71, 97, 101, 107, 109, 113, 127, 131, 149, 151, 173, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 257, 269, 281, 307, 311, 313, 317, 347, 349, 359, 373, 383, 389, 401, 409, 419, 421, 433, 439, 461, 479, 503, 509, 557, 563, 569, 571, 607 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The sequence consists of such odd prime numbers p that satisfy li[psi(p^3)]-li[psi(p^3-1)]<1/3, where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function. LINKS Dana Jacobsen, Table of n, a(n) for n = 1..314 M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011. MAPLE # The function PlanatSole(n, r) is in A196667. A196669 := n -> PlanatSole(n, 3); # Peter Luschny, Oct 23 2011 MATHEMATICA ChebyshevPsi[n_] := Log[LCM @@ Range[n]]; Reap[Do[If[LogIntegral[ChebyshevPsi[p^3]] - LogIntegral[ChebyshevPsi[p^3 - 1]] < 1/3, Print[p]; Sow[p]], {p, Prime[Range[2, 120]]}]][[2, 1]] (* Jean-François Alcover, Jul 14 2018, updated Dec 06 2018 *) PROG (Magma) Mangoldt:=function(n); if #Factorization(n) eq 1 then return Log(Factorization(n)[1][1]); else return 0; end if; end function; tcheb:=function(n); x:=0; for i in [1..n] do x:=x+Mangoldt(i); end for; return(x); end function; jump3:=function(n); x:=LogIntegral(tcheb(NthPrime(n)^3))-LogIntegral(tcheb(NthPrime(n)^3-1)); return x; end function; Set3:=[]; for i in [2..1000] do if jump3(i)-1/3 lt 0 then Set3:=Append(Set3, NthPrime(i)); NthPrime(i); end if; end for; Set3; (Sage) def A196669(n) : return PlanatSole(n, 3) # The function PlanatSole(n, r) is in A196667. # Peter Luschny, Oct 23 2011 (Perl) use ntheory ":all"; forprimes { say if 3 * (LogarithmicIntegral(chebyshev_psi(\$_**3)) - LogarithmicIntegral(chebyshev_psi(\$_**3-1))) < 1 } 3, 1000; # Dana Jacobsen, Dec 29 2015 CROSSREFS Cf. A196667, A196668, A196670. Sequence in context: A080821 A175275 A094517 * A322548 A049719 A155555 Adjacent sequences: A196666 A196667 A196668 * A196670 A196671 A196672 KEYWORD nonn AUTHOR Michel Planat, Oct 05 2011 EXTENSIONS Corrected and extended by Dana Jacobsen, Dec 29 2015 STATUS approved

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Last modified May 26 13:56 EDT 2024. Contains 372826 sequences. (Running on oeis4.)